Resonance, and the speeds a machine has to avoid
Natural frequency, and why some settings are forbidden.
Some machine speeds have to be avoided because running a machine so that its repeating disturbance (an unbalanced shaft turning, a mechanism cycling back and forth) lands exactly on the structure's natural frequency lets every cycle add a little more energy to the vibration. An ordinary push comes and goes, while a perfectly timed repeated push keeps building on itself.
Pushing a swing at the right moment
A small push given to a playground swing just as it reaches the top of its arc and starts back builds it higher and higher from very little effort, each push adding cleanly onto the momentum already there. Push at random moments, sometimes helping and sometimes fighting the swing's motion, and it barely gains height however much effort goes in, because the mistimed pushes cancel much of what the well-timed ones achieve. A single bad push among many good ones is easily absorbed. A sustained run of pushes landing at the wrong moment can bring the swing to a stop faster than letting it coast, so it is the consistency of the timing, far more than the strength of any one push, that decides whether reinforcement or cancellation wins.
A rotating machine whose imbalance disturbs its structure at exactly the natural frequency is doing what the well-timed push does, reinforcing its own vibration cycle after cycle until the amplitude is large enough to matter, sometimes large enough to damage the machine.
Timing, and why a tiny imbalance is enough
Every structure vibrates most easily at its natural frequency, the same frequency the previous article described as the ceiling on how fast a mechanism can be driven cleanly. Disturbed at any other frequency, a structure absorbs and dissipates the disturbance without building toward anything dramatic. At the natural frequency, each new push arrives just as the structure is already moving the way that push reinforces, adding a little energy every cycle instead of fighting the energy stored from the cycle before. The reinforcement depends on timing alone, which is why a tiny imbalance can build a large vibration purely by arriving at the right moment on every revolution.
Damping, and how far a resonance can grow
A real structure never builds vibration without limit. Friction, air resistance and internal material losses, together called damping, bleed away a share of the energy as heat on every cycle, and a resonating structure settles wherever the energy added by the disturbance and the energy bled away by damping balance. Some damping is engineered in, such as a rubber bush chosen for how much energy it absorbs each cycle. Some is simply inherent, like the quiet internal friction inside any metal part flexing back and forth.
The balance can sit a long way up. For a lightly damped structure, the peak vibration at resonance is roughly the static deflection divided by twice the damping ratio, so a welded steel frame with damping of about two percent of critical can swing some twenty-five times as far as the same force would bend it if held steady. A washing machine shaking hardest at one point while ramping up to full spin, then settling into a smoother spin once past it, shows this in any kitchen, the drum passing through its critical speed on the way to the faster, calmer one it was designed to run at.
Forbidden speeds, and more than one of them
Equipment specifications therefore often list forbidden speed ranges, to be passed through quickly during start-up and shutdown and never run at continuously. Designing rotating or cyclically loaded machinery means identifying the structure's natural frequencies deliberately, checking them against every speed the machine is expected to run at, and marking any speed too close to one as off limits for sustained running. Where a machine must operate continuously near a natural frequency it cannot avoid, enough damping has to be engineered into the structure to keep the amplitude within safe limits.
Most real structures have several natural frequencies at once, each belonging to a different way the structure can flex, and a design cleared against the first one an engineer checked can still resonate badly at a second or third nobody looked for. A thorough check covers the lowest few.
That closes this set roughly where it opened. Stiffness turns out to be a whole shape's worth of behaviour, different in every direction and at every frequency. A machine that feels solid, moves quickly and cleanly, and avoids every one of its resonances is one where that whole behaviour was understood, instead of one where a single rough stiffness figure was checked once and trusted for everything.